LMFDB - L-function 2-3200-40.29-c1-0-49

Dirichlet series

L(s) = 1

− 3·9^-s + 4·13^-s − 2i·17^-s − 4i·29^-s − 12·37^-s + 10·41^-s + 7·49^-s − 4·53^-s − 12i·61^-s − 6i·73^-s + 9·81^-s + 10·89^-s − 18i·97^-s − 20i·101^-s − 20i·109^-s + ⋯

L(s) = 1

− 9^-s + 1.10·13^-s − 0.485i·17^-s − 0.742i·29^-s − 1.97·37^-s + 1.56·41^-s + 49^-s − 0.549·53^-s − 1.53i·61^-s − 0.702i·73^-s + 81^-s + 1.05·89^-s − 1.82i·97^-s − 1.99i·101^-s − 1.91i·109^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3200$ = $2^{7} \cdot 5^{2}$
Sign:	$0.316 + 0.948i$
Analytic conductor:	$25.5521$
Root analytic conductor:	$5.05491$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3200} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3200,\ (\ :1/2),\ 0.316 + 0.948i)$

Particular Values

$L(1)$	$\approx$	$1.394487793$
$L(\frac12)$	$\approx$	$1.394487793$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 1 $
	5	$ 1 $
good	3	$ 1 + 3T^{2} $
	7	$ 1 - 7T^{2} $
	11	$ 1 - 11T^{2} $
	13	$ 1 - 4T + 13T^{2} $
	17	$ 1 + 2iT - 17T^{2} $
	19	$ 1 - 19T^{2} $
	23	$ 1 - 23T^{2} $
	29	$ 1 + 4iT - 29T^{2} $
	31	$ 1 + 31T^{2} $
	37	$ 1 + 12T + 37T^{2} $
	41	$ 1 - 10T + 41T^{2} $
	43	$ 1 + 43T^{2} $
	47	$ 1 - 47T^{2} $
	53	$ 1 + 4T + 53T^{2} $
	59	$ 1 - 59T^{2} $
	61	$ 1 + 12iT - 61T^{2} $
	67	$ 1 + 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 + 6iT - 73T^{2} $
	79	$ 1 + 79T^{2} $
	83	$ 1 + 83T^{2} $
	89	$ 1 - 10T + 89T^{2} $
	97	$ 1 + 18iT - 97T^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−8.580967856813785012333115446052, −7.87229092446838130368023295305, −7.01234917133769375689332390362, −6.12899400395768050817262831828, −5.61813737571389026957901027864, −4.67889714849769342898730033339, −3.68579418290606987770970817854, −2.94353723896191981835740042222, −1.86274877259394954740602944111, −0.48192076835468992320752258759, 1.08645371298573824041980669375, 2.29588520903813057044376099748, 3.31369348684660704816412464670, 3.98987411273445258550065834144, 5.12912111247385227923289009276, 5.80951898610700459269131207622, 6.45559539554391060175783871911, 7.34920351604880914188833212294, 8.218479137035831349038979197212, 8.789588214984616936995658760511

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.394487793\)
\(L(\frac12)\)	\(\approx\)	\(1.394487793\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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